Let $(X,\tau)$ be a topological space. I need to show that if every subset is closed then it is a discrete space.
For finite $X$, let $S$ be a subset of $X$. Since $S$ is closed, $X \setminus S \in \tau$. But $X \setminus S \subseteq X$. Therefore, $X \setminus S$ is closed and $S$ is open. Since the choice of $S$ was arbitrary, $(X,\tau)$ is a discrete space.
My doubt is whether the proof holds for $X$ being infinite too.